HEX OF MATHEMATICS metric functions had first presented' themselves this way. Would it not have been 'more natural' to consider the inverses of these functions, that is, the familiar trigonometric functions themselves as the given functions to be studied and analysed? Undoubtedly; but in shoals of more advanced problems, the simplest of which is that of finding the length of the arc of an ellipse by the integral calculus, the awkward inverse 'elliptic' (not 'circular', as for the arc of a circle) functions presented themselves first It took Abel to see that these functions should be 'inverted* and studied, precisely as in the case of sin x, cos aj instead of sin"1 x9 cos"1 «r. Simple, was it not? Yet Legendre, a great mathematician, spent more than forty years over his "elliptic integrals' (the awkward 'inverse functions* of his problem) without ever once suspecting that he should invert.* This extremely simple, uncommonsensical way of looking at an apparently simple but profoundly recondite problem was one of the greatest mathematical advances of the nineteenth century. All this however was but the beginning, although a suffi- cientlv tremendous beginning — like Kipling's dawn coming up like thunder - of what Abel did hi his magnificent theorem and in his work on elliptic functions. The trigonometric or circular functions have & single real period, thus sin (as + 2ir) = sin #, etc. Abel discovered that his new functions provided by the inversion of an elliptic integral have precisely two periods, whose ratio is imaginary. After that, Abel's followers in this direction - Jacobi, Rosenhain, Weierstrass, Riemann, and many more - mined deeply into Abel's great theorem and by carrying on and extending his ideas discovered functions of n variables having 2n periods. Abel himself carried the exploita- tion of his discoveries far. His successors have applied all this work to geometry, mechanics, parts of mathematical physics, and other tracts of mathematics, solving important problems * In ascribing priority to Abel, rather than 'joint discovery' to Abel and Jaoobi, in this matter, I have followed Mittag-Leffler. From a thorough acquaintance with all the published evidence, J am con- vinced that AbeFs claim is indisputable, although Jacobi's com- patriots argue otherwise. 35G